Ideal Gas in a Gravitational Field

1. The problem statement, all variables and given/known data
Im given an ideal Gas confined in an container with quadratic bottom area of size L^2 and open at the top.
The whole thing is set in V(z) = m*g*z surely and contains N Atoms at Temperature T.
Compute the Helmoltz Free Energy.

3. The attempt at a solution
My first idea was to plug in the barometric hight formula into the expression for p and integrate over the whole thing allways integration over volume V = L^2*dz so that i get some F(z+dz) = F(z)*dz and integrate the whole thing for z from 0 to infinity.
But im kinda stuck with the -S*dT part. Regarding the fact that T is allways constant it thould be -S*T ( and i think it surely is :) ) but how to compute S ? I honestly don´t got any idea. ( in fact i have my problems computing entropies so i would be nice if you give me some hint and not the solution right away i really wanna understand this stuff :).